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Formulation and multigrid solution of Cauchy-Riemann optimal control problems

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Computing and Visualization in Science

Abstract

The formulation of optimal control problems governed by Cauchy-Riemann equations is presented. A distributed control mechanism through divergence and curl sources is considered with the boundary conditions of mixed type. A Lagrange multiplier framework is introduced to characterize the solution to Cauchy-Riemann optimal control problems as the solution of an optimality system of four first-order partial differential equations and two optimality conditions. To solve the optimality system, staggered grids and multigrid methods are investigated. It results that staggered grids provide a natural collocation of the optimization variables and second-order accurate solutions are obtained. The proposed multigrid scheme is based on a coarsening by a factor of three that results in a nested hierarchy of staggered grids. On these grids a distributed-Gauss-Seidel and gradient-based smoothing scheme is employed. Results of numerical experiments validate the proposed optimal control formulation and demonstrate the effectiveness of the staggered-grids multigrid solution procedure.

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References

  1. Adams R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17, 35–92 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arian, E., Ta’asan, S.: Multigrid one shot methods for optimal control problems: infinite dimensional control. ICASE, Report No. 94-52 (1994)

  4. Borzì A., Kunisch K., Kwak D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41, 1477–1497 (2002)

    Article  MathSciNet  Google Scholar 

  5. Borzì A., Morton K.W., Süli E., Vanmaele M.: Multilevel solution of cell vertex Cauchy-Riemann equations. SIAM J. Sci. Comput. 18, 441–459 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borzì A., Schulz V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361–395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brandt, A., Dinar, N.: Multigrid solutions to elliptic flow problems. In: Numerical Methods for PDEs, pp. 53–147 (1979)

  8. Dendy J.E., Moulton J.D. Jr.: Black box multigrid with coarsening by a factor of three. Numer. Linear Algebra Appl. 17, 577–598 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, Rhode Island (2002)

  10. Fix G., Rose M.: A comparative study of finite element and finite difference methods for Cauchy-Riemann type equations. SIAM J. Numer. Anal. 22, 250–261 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ghil M., Balgovind R.: A fast Cauchy-Riemann solver. Math. Comp. 33, 585–635 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grisvard P.: Singularities in Boundary Value Problems. Springer, Berlin (1992)

    MATH  Google Scholar 

  13. Gunzburger M.D., Lee H.-C.: Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions. Appl. Math. Comput. 100, 49–70 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hackbusch W.: Multi-grid Methods and Applications. Springer, New York (1985)

    MATH  Google Scholar 

  15. Hackbusch W.: Elliptic Differential Equations. Springer, New York (1992)

    MATH  Google Scholar 

  16. Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications. SIAM, Philadelphia (2008)

  17. Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  18. Oosterlee C.W., Gaspar F.J.: Multigrid methods for the Stokes system. Comput. Sci. Eng. 8, 34–43 (2006)

    Article  Google Scholar 

  19. Sidilkover D.: Factorizable schemes for the equations of fluid flow. Appl. Numer. Math. 41, 423–436 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ta’asan, S.: Canonical forms of multidimensional steady inviscid flows, Report No. 93-34, ICASE, Virginia (1993)

  21. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. AMS, Chicago (2010)

  22. Trottenberg U., Oosterlee C., Schüller A.: Multigrid. Academic Press, London (2001)

    MATH  Google Scholar 

  23. Vanka S.F.: Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comp. Phys. 65, 138–158 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vanmaele M., Morton K.W., Süli E., Borzì A.: Analysis of the cell vertex finite volume method for the Cauchy-Riemann equations. SIAM J. Numer. Anal. 34, 2043–2062 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010, Graz, Austria

    M. M. Butt

  2. Department of Mathematics, GC University, Katchery Road, Lahore, 54000, Pakistan

    M. M. Butt

  3. Institut für Mathematik, Universität Würzburg, Hubland Campus Nord, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

    A. Borzì

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  1. M. M. Butt
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  2. A. Borzì
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Correspondence to A. Borzì.

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Communicated by C. W. Oosterlee and A. Borzi.

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Butt, M.M., Borzì, A. Formulation and multigrid solution of Cauchy-Riemann optimal control problems. Comput. Visual Sci. 14, 79–90 (2011). https://doi.org/10.1007/s00791-011-0161-9

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  • DOI: https://doi.org/10.1007/s00791-011-0161-9

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